[math-fun] Punishment for breaking the laws of math (was Re: Bug in my published 34 yr old formula !?)
The early 60s Friden mechanical calculator could be semi-destroyed by taking the square root of a 20-digit product when the carriage was too close to the wall. Hundreds of dollars when dollars were worth something. --rwg Flashback: This was at a Naval Air Development Center, divided into several large Labs. There was an Airborne Electronics Lab where they worked (rather unsuccessfully) on flat panel displays, a gyro navigation lab, and a lab with a human centrifuge where the Mercury Astronauts trained. There was also ACL, the "Airborne" Computer Lab. My lab was Antisubmarine Warfare Systems Analysis. About all we did was compute. On mechanical calculators! Because ACL had a political stranglehold forbidding the other labs from owning computers! If you needed a computation, you submitted it to ACL. And _months_ later, you got your pathetic results. Because they were a bunch of government fossils who only did plugboard analog! The stranglehold finally broke in 1962, when my lab got its first computer--a PB250 with delay-line memory, programmed in octal, with seven ways to get a quotient|remainder wrong, but a perfect square root instruction! It is outrageous that square root took decades to show up in mainstream instruction sets. Logic designers apparently failed to notice how simple the schoolboy algorithm is in binary. So it didn't get into C. And who would design an instruction not in C? On 2018-06-10 12:14, Keith F. Lynch wrote:
"David Wilson" <davidwwilson@comcast.net> wrote:
Tomas Rokicki wrote:
I suspect the laws of mathematics may have changed over those 34 years.
Or at least the fines and jail terms.
I once got in serious trouble for breaking the law against dividing by zero.
When I was a small child, my father let me visit his office and play with his electro-mechanical calculator. He warned me not to divide by zero, but I did it anyway. The machine started grinding away, and kept going and going. Pulling the plug made it stop, but it started again as soon as it was plugged in again. He told me it had to be sent back to the factory for repairs.
For youngsters who have no idea what I'm talking about, I found a video of someone doing the same, only on a machine that has an interrupt switch: https://www.youtube.com/watch?v=7Kd3R_RlXgc
For youngsters who have no idea what I'm talking about, I found a video of someone doing the same, only on a machine that has an interrupt switch: https://www.youtube.com/watch?v=7Kd3R_RlXgc
The Friden square root algorithm (the calculator was released in 1965) is a thing of beauty, and completely different from square root algorithms that are more common these days. The starting point is the sum of odd integers is successive squares, but it is far more sophisticated. Knowing that if 1+3+5+..+k=n^2 then 1+3+5+…+(10k_9)=100n^2, you can use a shift and subtract algorithm to find digits very quickly. It is very closely related to the ENIAC square root algorithm, and surprisingly, turns out to be equivalent to an algorithm Frazer Jarvis published in 2005, but with a completely different proof. It turns out that the same trick works in any base, not just ten, and is particularly nice base two. And is not the same as the digit by digit approach, even digit by digit, that we were taught in high school back in the day. I really need to get my book published on all this stuff :-) Steve -- Stephen Lucas, Professor Department of Mathematics and Statistics MSC 1911, James Madison University, Harrisonburg, VA 22807 USA Phone 540 568 5104, Fax 540 568 6857, Web http://educ.jmu.edu/~lucassk/ Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other) Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato) On Jun 10, 2018, at 9:06 PM, Bill Gosper <billgosper@gmail.com<mailto:billgosper@gmail.com>> wrote: The early 60s Friden mechanical calculator could be semi-destroyed by taking the square root of a 20-digit product when the carriage was too close to the wall. Hundreds of dollars when dollars were worth something. --rwg Flashback: This was at a Naval Air Development Center, divided into several large Labs. There was an Airborne Electronics Lab where they worked (rather unsuccessfully) on flat panel displays, a gyro navigation lab, and a lab with a human centrifuge where the Mercury Astronauts trained. There was also ACL, the "Airborne" Computer Lab. My lab was Antisubmarine Warfare Systems Analysis. About all we did was compute. On mechanical calculators! Because ACL had a political stranglehold forbidding the other labs from owning computers! If you needed a computation, you submitted it to ACL. And _months_ later, you got your pathetic results. Because they were a bunch of government fossils who only did plugboard analog! The stranglehold finally broke in 1962, when my lab got its first computer--a PB250 with delay-line memory, programmed in octal, with seven ways to get a quotient|remainder wrong, but a perfect square root instruction! It is outrageous that square root took decades to show up in mainstream instruction sets. Logic designers apparently failed to notice how simple the schoolboy algorithm is in binary. So it didn't get into C. And who would design an instruction not in C? On 2018-06-10 12:14, Keith F. Lynch wrote: "David Wilson" <davidwwilson@comcast.net<mailto:davidwwilson@comcast.net>> wrote: Tomas Rokicki wrote: I suspect the laws of mathematics may have changed over those 34 years. Or at least the fines and jail terms. I once got in serious trouble for breaking the law against dividing by zero. When I was a small child, my father let me visit his office and play with his electro-mechanical calculator. He warned me not to divide by zero, but I did it anyway. The machine started grinding away, and kept going and going. Pulling the plug made it stop, but it started again as soon as it was plugged in again. He told me it had to be sent back to the factory for repairs. For youngsters who have no idea what I'm talking about, I found a video of someone doing the same, only on a machine that has an interrupt switch: https://urldefense.proofpoint.com/v2/url?u=https-3A__www.youtube.com_watch-3... For youngsters who have no idea what I'm talking about, I found a video of someone doing the same, only on a machine that has an interrupt switch: https://urldefense.proofpoint.com/v2/url?u=https-3A__www.youtube.com_watch-3... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...
This reminds me that the rounded square root of an integer can be derived using only a truncating square root algorithm, and without conditionals or guess-then-test. That is, t := sqrtFloor(4n). answer := floor(t/2) + (t mod 2) If the calculations are performed with arbitrary precision integers, rounding is always correct. On 6/10/18 18:55 , Lucas, Stephen K - lucassk wrote:
The Friden square root algorithm (the calculator was released in 1965) is a thing of beauty, and completely different from square root algorithms that are more common these days. The starting point is the sum of odd integers is successive squares, but it is far more sophisticated. Knowing that if 1+3+5+..+k=n^2 then 1+3+5+…+(10k_9)=100n^2, you can use a shift and subtract algorithm to find digits very quickly. It is very closely related to the ENIAC square root algorithm, and surprisingly, turns out to be equivalent to an algorithm Frazer Jarvis published in 2005, but with a completely different proof.
It turns out that the same trick works in any base, not just ten, and is particularly nice base two. And is not the same as the digit by digit approach, even digit by digit, that we were taught in high school back in the day.
participants (3)
-
Andres Valloud -
Bill Gosper -
Lucas, Stephen K - lucassk